I need to verify that a function is a solution of a logistics DE

Verify by direct cauculation that if k, C, and d are constants, then the function P(t) = C/(1+d*e[tex]^{-kCt}[/tex]) is a solution of the logistic DE P' = kP(C-P).

2. Relevant equations

I don't think there are any for this problem. :)

3. The attempt at a solution

Okay, so ... uh ... I guess in this problem I should just be looking for the derivative of the original equation. So here goes ....

P(t) = C/(1+d*e[tex]^{-kCt}[/tex])
P(t) = C(1+d*e[tex]^{-kCt}[/tex])[tex]^{-1}[/tex] -- [I just moved the bottom part to the top.]
P(t) = -(e[tex]^{-t}[/tex])[tex]^{-2}[/tex]*-1 (chain rule) <-- I think this is where I go wrong. C, k, and d are constants so I just made their derivaties one. Is that the right thing to do? Because somehow I get the feeling that the third line of work here isn't going to get me to the answer.